# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

90,248 questions
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### If $X$ is a complex Banach space, is the set $T \in L(X)$ with finite dimensional kernel dense?

If $X$ is a complex Banach space, is the set $T \in L(X)$ with finite dimensional kernel dense? Here $L(X)$ is the set of bounded linear operators from $X$ to itself equipped with the norm topology. ...
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### Cantor's Theorem proof

I'm having problems understanding the Cantor's theorem proof. Can't we just assume $f$ is surjective, say the cardinality of $P(A)$ is greater than $A$ so by the pigeonhole principle there must be an ...
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### When are we not allowed to replaced $\sin{x}$ , $\tan{x}$… by $x$ in a limit where $x\to 0$?

In what situation, can we not replace for example $e^x$ with $x$ when when $x\to 0$, in a limit. Sorry if the question is extremely vague , English is not my native language. Thank you in advance
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### Can someone help me solve this limit? [on hold]

Can someone help me solve this limit? Thank you in advance. $$\lim_{x\to 0} \bigg( \frac{1+\tan{x}}{1+\sin{x}} \bigg)^{\frac{1}{\sin{x}}}$$ If possible without L'Hospital, the exercise gives more ...
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### Bijectiveness in a neighborhood of $(0,\pi/2)$

Question: Define $g: \mathbb{R^2}\to \mathbb{R^2}$ by $g(x,y)=(y\cos x,(x+y)\sin y)$. Show that g maps a neighborhood of $(0,\pi/2)$ bijectively to a neighborhood of $(\pi/2,\pi/2)$. What I did/...
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### how functional reduce to Y(x,y,y') = y(x) + e x n(x)

How this functional can be expressed as above linear function . what math concepts and topics clear this fully and meaningful mathematical explanation.
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### Trying to understand a standard example of weakly but not strongly measurable function

I consider quite a standard example of a function that is weakly measurable but is not strongly measurable. Unfortunatelly, I don't fully understand it. Let $X=l_2([0,1])$. Then $X$ equipped with a ...
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### What is the function you get when you double the arguments of sin and cos in the Fourier Series of another function?

Suppose $$f(x) =\frac{a_0}{2}+\sum_{k=1}^\infty a_k\cos(kx)+b_k\sin(kx)$$ Then what is $$S(x)=\frac{a_0}{2}+\sum_{k=1}^\infty a_k\cos(2kx)+b_k\sin(2kx)$$ in terms of $f(x)$. I tried writing down ...
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### Extension of domain of contactive semigroups, and why contractivity is important?

I am reading a paper, where I think the authors use this fact that the contractivity of a semigroup on a space of initial values, imeadiatly implies that we can extend the domain of semigroup to this ...
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### I am stuck somewhere in this analysis problem

Consider a real valued continuous function f on [0,1] such that f is differentiable on (0,1) and f(0)=f(1)=0. Does there exist some c in (0,1) where f(c)=f'(c)? It seems like the answer is yes. I am ...
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### Proof that continuous curve in $\mathbb{R}^2$ has Lebesgue measure zero

Suppose $\Gamma$ is a curve $y = f(x)$ in $\mathbb{R}^2$, where $f$ is continuous. Show that $m(\Gamma)=0.$ [Hint: Cover $\Gamma$ by rectangles, using the uniform continuity of $f$.] My attempt: ...
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### Prove that inequality $\sqrt{2\sqrt{4\sqrt{8…\sqrt{2^n}}}} \leqslant n+1$

Let $n$ be the integer. Prove that $$\sqrt{2\sqrt{4\sqrt{8....\sqrt{2^n}}}} \leqslant n+1$$ SOURCE: BANGLADESH MATH OLYMPIAD I am a new beginner at the infinite radical and sequence. I don't know ...
### When does a convex sequence $S_n$ satisfy $S_{a+b+c} - (S_{a+b}+S_{a+c}+S_{b+c})+S_a+S_b+S_c\geq 0$
Let $S_n$ be a sequence of integers with $n\geq 0$ such that $S_0=S_1=S_2=0$ (also assume $S_n$ to not be completely zero) and $$S_{n-1}+S_{n+1}\geq 2S_n$$ i.e. $S_n$ is convex. Now consider the ...